Would it be ok if i would write down an inverse Fourier transform for my second equation like this: $$\mathcal{G}(k) = \int\limits_{-\infty}^{\infty} \mathcal{F}(x) \, e^{-ikx} \, \mathrm{d} x$$
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71GAFeb 27 '13 at 22:48

Some say that there should be a factor of $1/\sqrt{2 \pi}$ there and not $1/(2 \pi)$ BUT i think that this is just because some use Fourier transform of a Gaussian and if we want to normalize Gauss function $g(x) = a \exp\left[ - \frac{(x-\mu)^2}{2\sigma^2}\right]$, we need to choose $a = 1/\sqrt{2 \pi}$. This is a bit messed up in my head. Could anyone help me clear this up?
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71GAFeb 27 '13 at 22:28

You can add a constant and the transform is still valid.
Normalization changes.
As Penrose states, if you choose the second form, you can normalize multiplying by (2pi)^-1/2 with the benefit of having a symmetry in the anti-transform (that is the transform and the anti-transform have the same form, the only thing that changes is the integration variable)